What if I'm trading in a purchased vehicle for a leased vehicle? Do I get a credit for the taxes I already paid on the purchase towards the lease? Or will the taxes be refunded? Or do I lose all the money that I paid in taxes and I have to pay taxes on the new lease just the same as I would have had to anyway...

Taxes are never refunded. It would go to the taxes charged on the new vehicle.

You don't specify your state, but different states do things differently. In CA, I don't think you can use the sales tax credit via trade it, at all.

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2016 M3 - on order for October 19th European Delivery
2013 ActiveHybrid 3 - sold
2012 M3 coupe - sold